Average

, median and mode of a set of 250 points. The black curve represents the theoretical distribution used to generate the points, with the gray histogram depicting the actual distribution. s of two numbers, a and b, constructed as chords on a semicircle. The arithmetic, geometric and harmonic means are sometimes referred to as the "Pythagorean means". The most commonly used definition of the average is the arithmetic mean, i.e. the sum divided by the count, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. However, other meanings are sometimes used depending on the context, which can lead to confusion; for instance, in teaching, "average" sometimes refers to "the three Ms": mean, median, and mode. The median, defined as the value in the center after sorting the group, is usually used as the average in situations where the data is skewed or has outliers, in order to focus on the main part of the group rather than the long tail. For example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people. The harmonic mean, defined as the reciprocal of the mean of the reciprocals, is used in a variety of situations involving rates or ratios, such as computing the average speed from multiple measurements taken over the same distance. Indeed, unlike an arithmetic mean or median of speeds, a harmonic mean of speeds will give the value of the constant speed that would cause one to travel the same distance in the same amount of time. The mode represents the most common value found in the group. It can be used when the data is categorical rather than numeric, when the frequency of each value is relevant (such as where a histogram, bar chart, or probability density function is being referenced), or to find a value that represents the majority of the group. Other statistics that can be used as an average include the mid-range, the quadratic mean or the geometric mean, but they are rarely referred to as "the average". These different quantities all estimate the central tendency of a group, with each having their advantages and issues. Mathematically, they can be thought as solving different variational problems. ==General properties==

All averages of a collection are somewhere within its bounding box (and so for real numbers, between its maximum and minimum). Therefore, if a collection consists entirely of the same value, any average of it is that value. Most averages are monotonic, i.e. moving a member of it in one direction causes the average to move in the same direction, or equivalently, if two collections of numbers A and B have the same number of elements, and they can be arranged such that each entry in A ≥ the corresponding entry in B, then the average of A ≥ the average of B. All commonly-used averages are linearly homogeneous, i.e. multiplying every value by the same scale factor multiplies the average by that same scale factor. Most averages remain identical when the list of items is permuted, i.e. the ordering does not matter. ==List of possible averages==

Even though perhaps not an average, the \tauth quantile (another summary statistic that generalizes the median) can similarly be expressed as a solution to the optimization problem :\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \max\big((1-\tau)(x_i - x),\, \tau(x - x_i)\big) = \underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \big(|x - x_i| + (1 - 2\tau)\,x\big), which aims to minimize the total tilted absolute value loss (or quantile loss or pinball loss). Other more sophisticated averages are: trimmore sophistiean, trimedian, and normalized mean, with their generalizations. In a more general fashion, one can create their own average metric using the generalized f-mean: : y = f^{-1}\left(\frac{1}{n}\left[f(x_1) + f(x_2) + \cdots + f(x_n)\right]\right) where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric mean is another, using f(x) = log x. However, this method for generating means is not general enough to capture all averages. A more general method ==Moving average==

Given a time series, such as daily stock market prices or yearly temperatures, people often want to create a smoother series. This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the moving average: one chooses a number n and creates a new series by taking the arithmetic mean of the first n values, then moving forward one place by dropping the oldest value and introducing a new value at the other end of the list, and so on. This is the simplest form of moving average. More complicated forms involve using a weighted average. The weighting can be used to enhance or suppress various periodic behaviors and there is extensive analysis of what weightings to use in the literature on filtering. In digital signal processing the term "moving average" is used even when the sum of the weights is not 1.0 (so the output series is a scaled version of the averages). The reason for this is that the analyst is usually interested only in the trend or the periodic behavior. ==History==

Origin The first recorded time that the arithmetic mean was extended from 2 to n cases for the use of estimation was in the sixteenth century. From the late sixteenth century onwards, it gradually became a common method to use for reducing errors of measurement in various areas. At the time, astronomers wanted to know a real value from noisy measurement, such as the position of a planet or the diameter of the moon. Using the mean of several measured values, scientists assumed that the errors add up to a relatively small number when compared to the total of all measured values. The method of taking the mean for reducing observation errors was mainly developed in astronomy. A possible precursor to the arithmetic mean is the mid-range (the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation. : In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total to see if it is already naturally present among the numbers in the row; and we will find that the property of being [one] ninth [of the sum] only belongs to the [arithmetic] mean itself... Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves. Marine damage is either particular average, which is borne only by the owner of the damaged property, or general average, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean". A second English usage, documented as early as 1674 and sometimes spelled "averish", is as the residue and second growth of field crops, which were considered suited to consumption by draught animals ("avers"). There is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term for a tenant's day labour obligation to a sheriff, probably anglicised from "avera" found in the English Domesday Book (1085). The Oxford English Dictionary, however, says that derivations from German hafen haven, and Arabic ʿawâr loss, damage, have been "quite disposed of" and the word has a Romance origin. == Averages as a rhetorical tool ==

Due to the aforementioned colloquial nature of the term "average", the term can be used to obfuscate the true meaning of data and suggest varying answers to questions based on the averaging method (most frequently arithmetic mean, median, or mode) used. In his article "Framed for Lying: Statistics as In/Artistic Proof", University of Pittsburgh faculty member Daniel Libertz comments that statistical information is frequently dismissed from rhetorical arguments for this reason. However, due to their persuasive power, averages and other statistical values should not be discarded completely, but instead used and interpreted with caution. Libertz invites us to engage critically not only with statistical information such as averages, but also with the language used to describe the data and its uses, saying: "If statistics rely on interpretation, rhetors should invite their audience to interpret rather than insist on an interpretation." In many cases, data and specific calculations are provided to help facilitate this audience-based interpretation. ==See also==